Question: Solve for $z$, $ -\dfrac{4z + 7}{5z + 1} = -\dfrac{6}{5z + 1} + \dfrac{6}{25z + 5} $
Answer: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5z + 1$ $5z + 1$ and $25z + 5$ The common denominator is $25z + 5$ To get $25z + 5$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{4z + 7}{5z + 1} \times \dfrac{5}{5} = -\dfrac{20z + 35}{25z + 5} $ To get $25z + 5$ in the denominator of the second term, multiply it by $\frac{5}{5}$ $ -\dfrac{6}{5z + 1} \times \dfrac{5}{5} = -\dfrac{30}{25z + 5} $ The denominator of the third term is already $25z + 5$ , so we don't need to change it. This give us: $ -\dfrac{20z + 35}{25z + 5} = -\dfrac{30}{25z + 5} + \dfrac{6}{25z + 5} $ If we multiply both sides of the equation by $25z + 5$ , we get: $ -20z - 35 = -30 + 6$ $ -20z - 35 = -24$ $ -20z = 11 $ $ z = -\dfrac{11}{20}$